Clock Hands Trisecting Face?Date: 04/05/99 at 13:28:39 From: Ken Logan Subject: Problem Solving (Time problems) Graduate Level Dear Dr. Math, I would appreciate your help on the following problem. I have asked the same question once already but have not heard a response from you. Question: At what time (if any) do the three hands of a clock trisect its face? Here is my solution: There are 60 1-minute spaces around a clock face. The rate of the second hand is 60 spaces per minute, the rate of the minute hand is 1 space per minute, and the rate of the hour hand is 1/12 space per minute. We begin by letting all three hands begin at the 12:00 position. We let Sh represent the number of spaces the hour hand moves until the hands trisect the face, Sm the number of spaces the minute hands move, and Ss the number of spaces the second hand moves. By the distance = rate x time formula we have Ss = 60T, Sm = T and Sh = (1/12)T. Notice that the time it takes each hand to be equal spaces apart is the same. Furthermore, Ss = Sm + 1200 and Ss = Sh + 14,400, and because Sm = T we have Ss = T + 1200. By substitution we have T + 1200 = 60T. Solving this equation for T we have T = 20 20/59 minutes. This is the number of minutes until the second and minute hands are 20 minutes apart. Where do I go from here? Do I take an entirely different approach? Thank you, Ken Logan Date: 04/05/99 at 16:53:44 From: Doctor Anthony Subject: Re: Problem Solving (Time problems) Graduate Level It looks as if the three hands never exactly trisect the clock face. In 12 hours Hour hand moves 360 degrees Minute hand moves 12 x 360 degrees Second hand moves 12 x 60 x 360 = 720 x 360 degrees and these are in the ratio 1 : 12 : 720 If a = the angle moved through by hour hand, then 12a = the angle moved through by the minute hand and 720a = the angle moved through by the second hand. So we require 12a - a = 120 11a = 120 so a = 10.90909.... degrees and 12a = 130.90909....... But 720a = 7854.5454..... subtracting whole revolutions from this we get 7854.5454 - 7560 = 294.5454... But 294.5454 - 130.90909 = 163.6363 which does NOT equal 120 degrees. If we instead have 12a - a = 240 11a = 240 a = 21.81818 and then 12a = 261.81818 and 720a = 15709.09091 and subtracting whole revolutions 15709.09091 - 15480 = 229.0909 and this is clearly not 120 degrees from 261.81818 So in both cases we cannot get the hands to exactly trisect the face. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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